The LRT statistic is \(\Lambda =2 \log L\big (\hat{\theta }_{f u l l}\big )-2 \log L\big (\hat{\theta }_{\text{reduced}}\big ),\)  where \(\log L\big (\hat{\theta }_{\text{full}}\big )\) 
is the likelihood function for the full model containing all the parameters, and \(\log L\big (\hat{\theta }_{\text{reduced}}\big )\)  is the likelihood function value for the reduced model under \(H_{0}\) . Under the normality assumption and certain other regularity conditions ([1]}), when \(H_0\)  is true, \(S^2\)  follows a central \(\chi ^{2}_{d f}\)  where \(\mathrm {df}\)  is the difference between the number of parameters in the full model and the reduced model. The null hypothesis, \(H_0\) , is rejected if the observed value of \(S^2\)  exceeds the \((1-\alpha )^{\text{th}}\)  quantile of \(\chi ^{2}_{d f}\) .
